The article defines semialgebraic and semianalytic functions via their graphs and develops the analytic geometry of such sets. It provides background showing that these classes are closely tied to analytic structure, but it does not state that every semialgebraic function is analytic on an open dense semialgebraic subset.

Drusvyatskiy and Lewis recall that "An important feature of semialgebraic sets is that they can be decomposed into analytic manifolds." They define a C^p-stratification of a semialgebraic set Q as "a finite partition of Q into disjoint semialgebraic C^p manifolds {M_i} (called strata) with the property that for each index i, the intersection of the closure of M_i with Q is the union of some M_j's." They further state that, for semialgebraic set-valued mappings satisfying dim gph F ≤ m, one can find finitely many **open semialgebraic sets U_i** and analytic semialgebraic single-valued mappings G_i^j : U_i → R^n such that F admits an analytic localization on each U_i. This shows analytic behavior on open semialgebraic pieces for certain semialgebraic mappings under dimension assumptions, but it is a Sard-type result for graphs of dimension ≤ range dimension, not a general statement about every semialgebraic function on [0,1]^n.

The paper gives formal definitions of semialgebraic sets and real analytic functions, and it treats them as foundational notions in real algebraic geometry and real analysis. It is relevant background, but it does not provide direct evidence for the specific open-dense-analytic claim.

Drusvyatskiy shows that "the subdifferential of any semi-algebraic extended-real-valued function on R^n has n-dimensional graph" and discusses consequences for optimization problems with semi-algebraic data. He repeatedly uses the fact that semi-algebraic sets admit finite stratifications into C^ω manifolds and that semi-algebraic functions are **C^1 on each stratum of a suitable stratification of their domain**. However, the paper does not state that such a function is real analytic on a single open dense semialgebraic subset of R^n; the differentiability/regularity is piecewise (by strata) and the strata themselves can have various dimensions, many of which are not open in R^n.

Bolte and Pauwels consider optimization problems with semialgebraic data and use tools from semialgebraic geometry. They recall that semialgebraic sets can be decomposed into analytic manifolds and state that for semialgebraic sets "dense" is equivalent to "full measure" and "topologically generic." They also cite stratification results guaranteeing that semialgebraic mappings can be made analytic on strata after passing to open dense subsets of parameter spaces in certain genericity theorems. However, their genericity statements concern typical parameters in optimization problems, not an arbitrary semialgebraic function on [0,1]^n; they do not assert that every such function is real analytic on an open dense semialgebraic subset of its domain.

Bierstone and Milman discuss semianalytic and subanalytic sets and their stratifications. They note that a semianalytic subset of R^n is one locally defined by finitely many real-analytic equalities and inequalities, and that subanalytic sets admit stratifications satisfying various regularity conditions (Whitney, Verdier, etc.). They introduce the notion of semicoherence and show that certain subanalytic sets admit stratifications such that local formal ideals are generated by finitely many subanalytically parametrized formal power series. This provides powerful analytic stratification results, but these are about the **structure of sets and singularities**, not about every semialgebraic function being analytic on an open dense subset; the regularity is again piecewise on strata, not a single open dense region of analyticity in the ambient cube.

Benedetti and Shiota survey triangulation and stratification theorems: "Triangulation theorems have been proved for sets of increasing order of generality (semianalytic, subanalytic, Whitney stratified, etc.). In semialgebraic geometry, one can triangulate semialgebraic sets and maps with semialgebraic homeomorphisms." They rely on the fact that semialgebraic sets admit C^ω or C^k stratifications and that semialgebraic functions can be made C^k on each simplex of a suitable triangulation. However, nothing here says that every semialgebraic function is globally real analytic on an open dense semialgebraic subset of its domain; rather, the regularity is piecewise on simplices or strata.

The authors define f : R^m → R^n to be semialgebraic if its graph is a semialgebraic subset of R^m × R^n and then develop representation theorems: "We have presented new representation theorems for semialgebraic functions along with a novel neural network architecture capable of representing all bounded semialgebraic functions." They emphasize piecewise polynomial/semialgebraic structure rather than analyticity; semialgebraic functions are allowed to be defined by polynomial equalities and inequalities with branching, and are generally only **piecewise analytic**. The paper treats semialgebraic functions as a broad class including many non-analytic examples (e.g., piecewise-polynomial with kinks), which indicates that one cannot expect global real-analytic regularity on an open dense subset in general.

Hironaka develops the theory of subanalytic sets and proves that such sets admit local finite stratifications by real analytic manifolds satisfying Whitney conditions. Semialgebraic sets are special cases of subanalytic sets. The paper establishes that subanalytic and semialgebraic sets have **analytic stratifications**, and that definable functions are typically real analytic along strata of suitable stratifications. However, Hironaka does not assert that every semialgebraic function on a cube is real analytic on an open dense semialgebraic subset of the cube; the analyticity is local along strata, and singular behavior can remain on lower-dimensional subsets.

This survey explains a standard result in the semialgebraic setting: semialgebraic sets admit stratifications, and on an open dense definable subset one gets the expected manifold behavior. It is useful background for the claim, but it does not state the claimed theorem about every semialgebraic function being real analytic on an open dense semialgebraic subset.

The document defines a Nash map as a map which is analytic and semialgebraic, and states that semialgebraic sets admit finite semialgebraic cell decompositions. This supports the broader framework in which semialgebraic objects can have analytic pieces, but it does not by itself assert the claim for arbitrary semialgebraic functions on the cube.

The paper defines a semialgebraic function as one whose graph is semialgebraic, and it introduces Nash functions as real analytic semialgebraic functions on open semialgebraic sets. It also states an extension theorem for arc-analytic semialgebraic functions, with the extended function being real analytic outside the Zariski closure of the original set.

In real algebraic geometry, semialgebraic functions are piecewise analytic after stratification, and there are cell decomposition/stratification theorems implying generic regularity on strata. This background is consistent with the claim’s spirit, but it is not a direct citation from the supplied sources.